Properties

Label 4009.2.a.c.1.31
Level $4009$
Weight $2$
Character 4009.1
Self dual yes
Analytic conductor $32.012$
Analytic rank $1$
Dimension $71$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4009,2,Mod(1,4009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.0120261703\)
Analytic rank: \(1\)
Dimension: \(71\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.31
Character \(\chi\) \(=\) 4009.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.893036 q^{2} -2.08830 q^{3} -1.20249 q^{4} -3.63561 q^{5} +1.86492 q^{6} -2.58906 q^{7} +2.85994 q^{8} +1.36098 q^{9} +O(q^{10})\) \(q-0.893036 q^{2} -2.08830 q^{3} -1.20249 q^{4} -3.63561 q^{5} +1.86492 q^{6} -2.58906 q^{7} +2.85994 q^{8} +1.36098 q^{9} +3.24673 q^{10} -1.15709 q^{11} +2.51115 q^{12} -5.25839 q^{13} +2.31212 q^{14} +7.59223 q^{15} -0.149054 q^{16} -4.71801 q^{17} -1.21540 q^{18} +1.00000 q^{19} +4.37177 q^{20} +5.40672 q^{21} +1.03332 q^{22} -5.80291 q^{23} -5.97239 q^{24} +8.21767 q^{25} +4.69593 q^{26} +3.42276 q^{27} +3.11331 q^{28} -9.27186 q^{29} -6.78014 q^{30} -1.62332 q^{31} -5.58676 q^{32} +2.41634 q^{33} +4.21335 q^{34} +9.41281 q^{35} -1.63656 q^{36} +2.08729 q^{37} -0.893036 q^{38} +10.9811 q^{39} -10.3976 q^{40} +3.49915 q^{41} -4.82840 q^{42} +7.33492 q^{43} +1.39138 q^{44} -4.94798 q^{45} +5.18221 q^{46} +5.98392 q^{47} +0.311269 q^{48} -0.296775 q^{49} -7.33867 q^{50} +9.85259 q^{51} +6.32314 q^{52} +5.97254 q^{53} -3.05665 q^{54} +4.20672 q^{55} -7.40454 q^{56} -2.08830 q^{57} +8.28011 q^{58} +9.14132 q^{59} -9.12955 q^{60} -5.99586 q^{61} +1.44968 q^{62} -3.52365 q^{63} +5.28729 q^{64} +19.1175 q^{65} -2.15788 q^{66} +3.69923 q^{67} +5.67334 q^{68} +12.1182 q^{69} -8.40598 q^{70} -1.78491 q^{71} +3.89231 q^{72} -5.50261 q^{73} -1.86402 q^{74} -17.1609 q^{75} -1.20249 q^{76} +2.99577 q^{77} -9.80649 q^{78} +4.42549 q^{79} +0.541902 q^{80} -11.2307 q^{81} -3.12487 q^{82} +14.7807 q^{83} -6.50151 q^{84} +17.1528 q^{85} -6.55035 q^{86} +19.3624 q^{87} -3.30919 q^{88} +1.50034 q^{89} +4.41873 q^{90} +13.6143 q^{91} +6.97792 q^{92} +3.38996 q^{93} -5.34385 q^{94} -3.63561 q^{95} +11.6668 q^{96} +10.2071 q^{97} +0.265031 q^{98} -1.57477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.893036 −0.631472 −0.315736 0.948847i \(-0.602251\pi\)
−0.315736 + 0.948847i \(0.602251\pi\)
\(3\) −2.08830 −1.20568 −0.602839 0.797863i \(-0.705964\pi\)
−0.602839 + 0.797863i \(0.705964\pi\)
\(4\) −1.20249 −0.601243
\(5\) −3.63561 −1.62589 −0.812947 0.582337i \(-0.802138\pi\)
−0.812947 + 0.582337i \(0.802138\pi\)
\(6\) 1.86492 0.761352
\(7\) −2.58906 −0.978572 −0.489286 0.872123i \(-0.662743\pi\)
−0.489286 + 0.872123i \(0.662743\pi\)
\(8\) 2.85994 1.01114
\(9\) 1.36098 0.453659
\(10\) 3.24673 1.02671
\(11\) −1.15709 −0.348875 −0.174437 0.984668i \(-0.555811\pi\)
−0.174437 + 0.984668i \(0.555811\pi\)
\(12\) 2.51115 0.724906
\(13\) −5.25839 −1.45841 −0.729207 0.684293i \(-0.760111\pi\)
−0.729207 + 0.684293i \(0.760111\pi\)
\(14\) 2.31212 0.617941
\(15\) 7.59223 1.96031
\(16\) −0.149054 −0.0372635
\(17\) −4.71801 −1.14428 −0.572142 0.820154i \(-0.693887\pi\)
−0.572142 + 0.820154i \(0.693887\pi\)
\(18\) −1.21540 −0.286473
\(19\) 1.00000 0.229416
\(20\) 4.37177 0.977558
\(21\) 5.40672 1.17984
\(22\) 1.03332 0.220305
\(23\) −5.80291 −1.20999 −0.604996 0.796229i \(-0.706825\pi\)
−0.604996 + 0.796229i \(0.706825\pi\)
\(24\) −5.97239 −1.21911
\(25\) 8.21767 1.64353
\(26\) 4.69593 0.920948
\(27\) 3.42276 0.658711
\(28\) 3.11331 0.588360
\(29\) −9.27186 −1.72174 −0.860871 0.508823i \(-0.830081\pi\)
−0.860871 + 0.508823i \(0.830081\pi\)
\(30\) −6.78014 −1.23788
\(31\) −1.62332 −0.291556 −0.145778 0.989317i \(-0.546569\pi\)
−0.145778 + 0.989317i \(0.546569\pi\)
\(32\) −5.58676 −0.987609
\(33\) 2.41634 0.420631
\(34\) 4.21335 0.722584
\(35\) 9.41281 1.59106
\(36\) −1.63656 −0.272759
\(37\) 2.08729 0.343148 0.171574 0.985171i \(-0.445115\pi\)
0.171574 + 0.985171i \(0.445115\pi\)
\(38\) −0.893036 −0.144870
\(39\) 10.9811 1.75838
\(40\) −10.3976 −1.64401
\(41\) 3.49915 0.546475 0.273238 0.961947i \(-0.411905\pi\)
0.273238 + 0.961947i \(0.411905\pi\)
\(42\) −4.82840 −0.745038
\(43\) 7.33492 1.11856 0.559282 0.828977i \(-0.311077\pi\)
0.559282 + 0.828977i \(0.311077\pi\)
\(44\) 1.39138 0.209759
\(45\) −4.94798 −0.737602
\(46\) 5.18221 0.764075
\(47\) 5.98392 0.872844 0.436422 0.899742i \(-0.356246\pi\)
0.436422 + 0.899742i \(0.356246\pi\)
\(48\) 0.311269 0.0449278
\(49\) −0.296775 −0.0423964
\(50\) −7.33867 −1.03785
\(51\) 9.85259 1.37964
\(52\) 6.32314 0.876862
\(53\) 5.97254 0.820392 0.410196 0.911997i \(-0.365460\pi\)
0.410196 + 0.911997i \(0.365460\pi\)
\(54\) −3.05665 −0.415958
\(55\) 4.20672 0.567234
\(56\) −7.40454 −0.989474
\(57\) −2.08830 −0.276601
\(58\) 8.28011 1.08723
\(59\) 9.14132 1.19010 0.595049 0.803689i \(-0.297132\pi\)
0.595049 + 0.803689i \(0.297132\pi\)
\(60\) −9.12955 −1.17862
\(61\) −5.99586 −0.767692 −0.383846 0.923397i \(-0.625401\pi\)
−0.383846 + 0.923397i \(0.625401\pi\)
\(62\) 1.44968 0.184110
\(63\) −3.52365 −0.443938
\(64\) 5.28729 0.660911
\(65\) 19.1175 2.37123
\(66\) −2.15788 −0.265616
\(67\) 3.69923 0.451933 0.225966 0.974135i \(-0.427446\pi\)
0.225966 + 0.974135i \(0.427446\pi\)
\(68\) 5.67334 0.687993
\(69\) 12.1182 1.45886
\(70\) −8.40598 −1.00471
\(71\) −1.78491 −0.211830 −0.105915 0.994375i \(-0.533777\pi\)
−0.105915 + 0.994375i \(0.533777\pi\)
\(72\) 3.89231 0.458713
\(73\) −5.50261 −0.644032 −0.322016 0.946734i \(-0.604360\pi\)
−0.322016 + 0.946734i \(0.604360\pi\)
\(74\) −1.86402 −0.216688
\(75\) −17.1609 −1.98157
\(76\) −1.20249 −0.137935
\(77\) 2.99577 0.341399
\(78\) −9.80649 −1.11037
\(79\) 4.42549 0.497907 0.248953 0.968515i \(-0.419913\pi\)
0.248953 + 0.968515i \(0.419913\pi\)
\(80\) 0.541902 0.0605865
\(81\) −11.2307 −1.24785
\(82\) −3.12487 −0.345084
\(83\) 14.7807 1.62240 0.811199 0.584770i \(-0.198815\pi\)
0.811199 + 0.584770i \(0.198815\pi\)
\(84\) −6.50151 −0.709372
\(85\) 17.1528 1.86049
\(86\) −6.55035 −0.706342
\(87\) 19.3624 2.07587
\(88\) −3.30919 −0.352761
\(89\) 1.50034 0.159036 0.0795178 0.996833i \(-0.474662\pi\)
0.0795178 + 0.996833i \(0.474662\pi\)
\(90\) 4.41873 0.465775
\(91\) 13.6143 1.42716
\(92\) 6.97792 0.727499
\(93\) 3.38996 0.351523
\(94\) −5.34385 −0.551176
\(95\) −3.63561 −0.373006
\(96\) 11.6668 1.19074
\(97\) 10.2071 1.03638 0.518188 0.855267i \(-0.326607\pi\)
0.518188 + 0.855267i \(0.326607\pi\)
\(98\) 0.265031 0.0267722
\(99\) −1.57477 −0.158270
\(100\) −9.88163 −0.988163
\(101\) −9.67000 −0.962201 −0.481101 0.876665i \(-0.659763\pi\)
−0.481101 + 0.876665i \(0.659763\pi\)
\(102\) −8.79872 −0.871203
\(103\) 14.8149 1.45976 0.729879 0.683576i \(-0.239576\pi\)
0.729879 + 0.683576i \(0.239576\pi\)
\(104\) −15.0387 −1.47466
\(105\) −19.6567 −1.91830
\(106\) −5.33370 −0.518054
\(107\) −13.5153 −1.30657 −0.653285 0.757112i \(-0.726610\pi\)
−0.653285 + 0.757112i \(0.726610\pi\)
\(108\) −4.11583 −0.396046
\(109\) 12.1046 1.15941 0.579704 0.814827i \(-0.303168\pi\)
0.579704 + 0.814827i \(0.303168\pi\)
\(110\) −3.75675 −0.358192
\(111\) −4.35887 −0.413726
\(112\) 0.385909 0.0364650
\(113\) −8.28441 −0.779332 −0.389666 0.920956i \(-0.627410\pi\)
−0.389666 + 0.920956i \(0.627410\pi\)
\(114\) 1.86492 0.174666
\(115\) 21.0971 1.96732
\(116\) 11.1493 1.03519
\(117\) −7.15655 −0.661623
\(118\) −8.16353 −0.751514
\(119\) 12.2152 1.11976
\(120\) 21.7133 1.98214
\(121\) −9.66115 −0.878286
\(122\) 5.35452 0.484776
\(123\) −7.30726 −0.658873
\(124\) 1.95202 0.175296
\(125\) −11.6982 −1.04632
\(126\) 3.14675 0.280335
\(127\) −11.1304 −0.987664 −0.493832 0.869557i \(-0.664404\pi\)
−0.493832 + 0.869557i \(0.664404\pi\)
\(128\) 6.45178 0.570262
\(129\) −15.3175 −1.34863
\(130\) −17.0726 −1.49736
\(131\) −10.8833 −0.950879 −0.475440 0.879748i \(-0.657711\pi\)
−0.475440 + 0.879748i \(0.657711\pi\)
\(132\) −2.90561 −0.252901
\(133\) −2.58906 −0.224500
\(134\) −3.30354 −0.285383
\(135\) −12.4438 −1.07099
\(136\) −13.4932 −1.15703
\(137\) −1.22392 −0.104567 −0.0522833 0.998632i \(-0.516650\pi\)
−0.0522833 + 0.998632i \(0.516650\pi\)
\(138\) −10.8220 −0.921229
\(139\) −13.2395 −1.12296 −0.561479 0.827491i \(-0.689768\pi\)
−0.561479 + 0.827491i \(0.689768\pi\)
\(140\) −11.3188 −0.956611
\(141\) −12.4962 −1.05237
\(142\) 1.59399 0.133765
\(143\) 6.08441 0.508804
\(144\) −0.202859 −0.0169049
\(145\) 33.7089 2.79937
\(146\) 4.91403 0.406688
\(147\) 0.619754 0.0511164
\(148\) −2.50994 −0.206315
\(149\) −2.76637 −0.226630 −0.113315 0.993559i \(-0.536147\pi\)
−0.113315 + 0.993559i \(0.536147\pi\)
\(150\) 15.3253 1.25131
\(151\) −0.300056 −0.0244182 −0.0122091 0.999925i \(-0.503886\pi\)
−0.0122091 + 0.999925i \(0.503886\pi\)
\(152\) 2.85994 0.231971
\(153\) −6.42110 −0.519115
\(154\) −2.67533 −0.215584
\(155\) 5.90175 0.474040
\(156\) −13.2046 −1.05721
\(157\) 15.5334 1.23970 0.619849 0.784721i \(-0.287194\pi\)
0.619849 + 0.784721i \(0.287194\pi\)
\(158\) −3.95212 −0.314414
\(159\) −12.4724 −0.989128
\(160\) 20.3113 1.60575
\(161\) 15.0241 1.18406
\(162\) 10.0294 0.787984
\(163\) −5.38699 −0.421942 −0.210971 0.977492i \(-0.567663\pi\)
−0.210971 + 0.977492i \(0.567663\pi\)
\(164\) −4.20768 −0.328564
\(165\) −8.78487 −0.683901
\(166\) −13.1997 −1.02450
\(167\) 16.4343 1.27173 0.635863 0.771802i \(-0.280644\pi\)
0.635863 + 0.771802i \(0.280644\pi\)
\(168\) 15.4629 1.19299
\(169\) 14.6507 1.12697
\(170\) −15.3181 −1.17484
\(171\) 1.36098 0.104077
\(172\) −8.82014 −0.672529
\(173\) 18.1456 1.37959 0.689794 0.724006i \(-0.257701\pi\)
0.689794 + 0.724006i \(0.257701\pi\)
\(174\) −17.2913 −1.31085
\(175\) −21.2760 −1.60832
\(176\) 0.172468 0.0130003
\(177\) −19.0898 −1.43488
\(178\) −1.33986 −0.100426
\(179\) −6.32716 −0.472914 −0.236457 0.971642i \(-0.575986\pi\)
−0.236457 + 0.971642i \(0.575986\pi\)
\(180\) 5.94988 0.443478
\(181\) −7.44417 −0.553321 −0.276660 0.960968i \(-0.589228\pi\)
−0.276660 + 0.960968i \(0.589228\pi\)
\(182\) −12.1580 −0.901214
\(183\) 12.5211 0.925589
\(184\) −16.5960 −1.22347
\(185\) −7.58857 −0.557923
\(186\) −3.02736 −0.221977
\(187\) 5.45914 0.399212
\(188\) −7.19558 −0.524791
\(189\) −8.86174 −0.644596
\(190\) 3.24673 0.235543
\(191\) −24.3366 −1.76093 −0.880467 0.474107i \(-0.842771\pi\)
−0.880467 + 0.474107i \(0.842771\pi\)
\(192\) −11.0414 −0.796846
\(193\) 24.4987 1.76345 0.881726 0.471761i \(-0.156381\pi\)
0.881726 + 0.471761i \(0.156381\pi\)
\(194\) −9.11532 −0.654442
\(195\) −39.9229 −2.85894
\(196\) 0.356868 0.0254906
\(197\) 3.27248 0.233154 0.116577 0.993182i \(-0.462808\pi\)
0.116577 + 0.993182i \(0.462808\pi\)
\(198\) 1.40633 0.0999432
\(199\) −20.7297 −1.46949 −0.734743 0.678346i \(-0.762697\pi\)
−0.734743 + 0.678346i \(0.762697\pi\)
\(200\) 23.5020 1.66184
\(201\) −7.72508 −0.544885
\(202\) 8.63566 0.607603
\(203\) 24.0054 1.68485
\(204\) −11.8476 −0.829498
\(205\) −12.7215 −0.888511
\(206\) −13.2303 −0.921796
\(207\) −7.89763 −0.548923
\(208\) 0.783784 0.0543456
\(209\) −1.15709 −0.0800374
\(210\) 17.5542 1.21135
\(211\) 1.00000 0.0688428
\(212\) −7.18190 −0.493255
\(213\) 3.72742 0.255398
\(214\) 12.0696 0.825063
\(215\) −26.6669 −1.81867
\(216\) 9.78889 0.666049
\(217\) 4.20286 0.285309
\(218\) −10.8098 −0.732134
\(219\) 11.4911 0.776495
\(220\) −5.05852 −0.341045
\(221\) 24.8091 1.66884
\(222\) 3.89263 0.261256
\(223\) 14.8445 0.994064 0.497032 0.867732i \(-0.334423\pi\)
0.497032 + 0.867732i \(0.334423\pi\)
\(224\) 14.4645 0.966447
\(225\) 11.1841 0.745604
\(226\) 7.39828 0.492126
\(227\) −22.9605 −1.52394 −0.761972 0.647610i \(-0.775769\pi\)
−0.761972 + 0.647610i \(0.775769\pi\)
\(228\) 2.51115 0.166305
\(229\) −20.3046 −1.34176 −0.670882 0.741564i \(-0.734084\pi\)
−0.670882 + 0.741564i \(0.734084\pi\)
\(230\) −18.8405 −1.24231
\(231\) −6.25604 −0.411617
\(232\) −26.5169 −1.74092
\(233\) −3.19355 −0.209217 −0.104608 0.994513i \(-0.533359\pi\)
−0.104608 + 0.994513i \(0.533359\pi\)
\(234\) 6.39106 0.417796
\(235\) −21.7552 −1.41915
\(236\) −10.9923 −0.715539
\(237\) −9.24173 −0.600315
\(238\) −10.9086 −0.707100
\(239\) 10.0855 0.652374 0.326187 0.945305i \(-0.394236\pi\)
0.326187 + 0.945305i \(0.394236\pi\)
\(240\) −1.13165 −0.0730478
\(241\) −8.46605 −0.545346 −0.272673 0.962107i \(-0.587908\pi\)
−0.272673 + 0.962107i \(0.587908\pi\)
\(242\) 8.62776 0.554613
\(243\) 13.1847 0.845797
\(244\) 7.20994 0.461569
\(245\) 1.07896 0.0689321
\(246\) 6.52564 0.416060
\(247\) −5.25839 −0.334583
\(248\) −4.64258 −0.294804
\(249\) −30.8666 −1.95609
\(250\) 10.4469 0.660720
\(251\) 8.08792 0.510505 0.255253 0.966874i \(-0.417841\pi\)
0.255253 + 0.966874i \(0.417841\pi\)
\(252\) 4.23714 0.266915
\(253\) 6.71447 0.422135
\(254\) 9.93985 0.623682
\(255\) −35.8202 −2.24315
\(256\) −16.3363 −1.02102
\(257\) −18.9026 −1.17911 −0.589556 0.807728i \(-0.700697\pi\)
−0.589556 + 0.807728i \(0.700697\pi\)
\(258\) 13.6791 0.851621
\(259\) −5.40411 −0.335795
\(260\) −22.9885 −1.42568
\(261\) −12.6188 −0.781084
\(262\) 9.71919 0.600454
\(263\) 0.398805 0.0245914 0.0122957 0.999924i \(-0.496086\pi\)
0.0122957 + 0.999924i \(0.496086\pi\)
\(264\) 6.91058 0.425317
\(265\) −21.7138 −1.33387
\(266\) 2.31212 0.141765
\(267\) −3.13315 −0.191746
\(268\) −4.44827 −0.271721
\(269\) 10.6158 0.647255 0.323627 0.946185i \(-0.395098\pi\)
0.323627 + 0.946185i \(0.395098\pi\)
\(270\) 11.1128 0.676303
\(271\) 13.7579 0.835731 0.417865 0.908509i \(-0.362778\pi\)
0.417865 + 0.908509i \(0.362778\pi\)
\(272\) 0.703237 0.0426400
\(273\) −28.4306 −1.72070
\(274\) 1.09301 0.0660309
\(275\) −9.50855 −0.573387
\(276\) −14.5720 −0.877129
\(277\) −14.5666 −0.875223 −0.437612 0.899164i \(-0.644176\pi\)
−0.437612 + 0.899164i \(0.644176\pi\)
\(278\) 11.8233 0.709117
\(279\) −2.20930 −0.132267
\(280\) 26.9200 1.60878
\(281\) 18.5659 1.10755 0.553773 0.832667i \(-0.313187\pi\)
0.553773 + 0.832667i \(0.313187\pi\)
\(282\) 11.1595 0.664541
\(283\) 3.31199 0.196877 0.0984386 0.995143i \(-0.468615\pi\)
0.0984386 + 0.995143i \(0.468615\pi\)
\(284\) 2.14633 0.127361
\(285\) 7.59223 0.449725
\(286\) −5.43360 −0.321296
\(287\) −9.05950 −0.534765
\(288\) −7.60346 −0.448038
\(289\) 5.25958 0.309387
\(290\) −30.1033 −1.76772
\(291\) −21.3155 −1.24953
\(292\) 6.61681 0.387220
\(293\) 5.43984 0.317799 0.158899 0.987295i \(-0.449205\pi\)
0.158899 + 0.987295i \(0.449205\pi\)
\(294\) −0.553463 −0.0322786
\(295\) −33.2343 −1.93498
\(296\) 5.96951 0.346971
\(297\) −3.96043 −0.229808
\(298\) 2.47047 0.143111
\(299\) 30.5140 1.76467
\(300\) 20.6358 1.19141
\(301\) −18.9905 −1.09460
\(302\) 0.267961 0.0154194
\(303\) 20.1938 1.16010
\(304\) −0.149054 −0.00854883
\(305\) 21.7986 1.24819
\(306\) 5.73427 0.327807
\(307\) 23.0389 1.31490 0.657449 0.753499i \(-0.271635\pi\)
0.657449 + 0.753499i \(0.271635\pi\)
\(308\) −3.60237 −0.205264
\(309\) −30.9379 −1.76000
\(310\) −5.27047 −0.299343
\(311\) −6.99870 −0.396860 −0.198430 0.980115i \(-0.563584\pi\)
−0.198430 + 0.980115i \(0.563584\pi\)
\(312\) 31.4052 1.77797
\(313\) −23.2253 −1.31277 −0.656385 0.754426i \(-0.727915\pi\)
−0.656385 + 0.754426i \(0.727915\pi\)
\(314\) −13.8719 −0.782835
\(315\) 12.8106 0.721797
\(316\) −5.32159 −0.299363
\(317\) 10.2293 0.574533 0.287267 0.957851i \(-0.407253\pi\)
0.287267 + 0.957851i \(0.407253\pi\)
\(318\) 11.1383 0.624607
\(319\) 10.7284 0.600672
\(320\) −19.2225 −1.07457
\(321\) 28.2239 1.57530
\(322\) −13.4170 −0.747703
\(323\) −4.71801 −0.262517
\(324\) 13.5047 0.750263
\(325\) −43.2117 −2.39695
\(326\) 4.81078 0.266444
\(327\) −25.2779 −1.39787
\(328\) 10.0073 0.552563
\(329\) −15.4927 −0.854141
\(330\) 7.84521 0.431864
\(331\) 4.46663 0.245508 0.122754 0.992437i \(-0.460827\pi\)
0.122754 + 0.992437i \(0.460827\pi\)
\(332\) −17.7736 −0.975456
\(333\) 2.84075 0.155672
\(334\) −14.6764 −0.803059
\(335\) −13.4490 −0.734795
\(336\) −0.805893 −0.0439651
\(337\) −29.5396 −1.60913 −0.804563 0.593868i \(-0.797600\pi\)
−0.804563 + 0.593868i \(0.797600\pi\)
\(338\) −13.0836 −0.711652
\(339\) 17.3003 0.939623
\(340\) −20.6260 −1.11860
\(341\) 1.87832 0.101717
\(342\) −1.21540 −0.0657214
\(343\) 18.8918 1.02006
\(344\) 20.9774 1.13103
\(345\) −44.0570 −2.37195
\(346\) −16.2047 −0.871171
\(347\) −27.4734 −1.47485 −0.737424 0.675430i \(-0.763958\pi\)
−0.737424 + 0.675430i \(0.763958\pi\)
\(348\) −23.2830 −1.24810
\(349\) 11.6487 0.623539 0.311769 0.950158i \(-0.399078\pi\)
0.311769 + 0.950158i \(0.399078\pi\)
\(350\) 19.0003 1.01561
\(351\) −17.9982 −0.960674
\(352\) 6.46437 0.344552
\(353\) 36.7989 1.95861 0.979303 0.202399i \(-0.0648737\pi\)
0.979303 + 0.202399i \(0.0648737\pi\)
\(354\) 17.0479 0.906084
\(355\) 6.48923 0.344413
\(356\) −1.80414 −0.0956190
\(357\) −25.5089 −1.35008
\(358\) 5.65038 0.298632
\(359\) −25.9133 −1.36765 −0.683827 0.729644i \(-0.739686\pi\)
−0.683827 + 0.729644i \(0.739686\pi\)
\(360\) −14.1509 −0.745819
\(361\) 1.00000 0.0526316
\(362\) 6.64791 0.349406
\(363\) 20.1753 1.05893
\(364\) −16.3710 −0.858073
\(365\) 20.0053 1.04713
\(366\) −11.1818 −0.584483
\(367\) 25.8156 1.34756 0.673782 0.738930i \(-0.264669\pi\)
0.673782 + 0.738930i \(0.264669\pi\)
\(368\) 0.864947 0.0450885
\(369\) 4.76226 0.247913
\(370\) 6.77687 0.352312
\(371\) −15.4633 −0.802812
\(372\) −4.07638 −0.211351
\(373\) −9.65684 −0.500012 −0.250006 0.968244i \(-0.580433\pi\)
−0.250006 + 0.968244i \(0.580433\pi\)
\(374\) −4.87521 −0.252091
\(375\) 24.4293 1.26152
\(376\) 17.1136 0.882567
\(377\) 48.7551 2.51101
\(378\) 7.91385 0.407045
\(379\) 24.5808 1.26263 0.631315 0.775527i \(-0.282516\pi\)
0.631315 + 0.775527i \(0.282516\pi\)
\(380\) 4.37177 0.224267
\(381\) 23.2436 1.19080
\(382\) 21.7335 1.11198
\(383\) −18.0944 −0.924580 −0.462290 0.886729i \(-0.652972\pi\)
−0.462290 + 0.886729i \(0.652972\pi\)
\(384\) −13.4732 −0.687553
\(385\) −10.8914 −0.555079
\(386\) −21.8782 −1.11357
\(387\) 9.98266 0.507447
\(388\) −12.2739 −0.623113
\(389\) −4.69382 −0.237986 −0.118993 0.992895i \(-0.537967\pi\)
−0.118993 + 0.992895i \(0.537967\pi\)
\(390\) 35.6526 1.80534
\(391\) 27.3782 1.38457
\(392\) −0.848758 −0.0428687
\(393\) 22.7276 1.14645
\(394\) −2.92244 −0.147230
\(395\) −16.0894 −0.809544
\(396\) 1.89364 0.0951589
\(397\) 8.52553 0.427884 0.213942 0.976846i \(-0.431370\pi\)
0.213942 + 0.976846i \(0.431370\pi\)
\(398\) 18.5123 0.927939
\(399\) 5.40672 0.270675
\(400\) −1.22488 −0.0612438
\(401\) 15.1496 0.756533 0.378266 0.925697i \(-0.376520\pi\)
0.378266 + 0.925697i \(0.376520\pi\)
\(402\) 6.89878 0.344080
\(403\) 8.53603 0.425210
\(404\) 11.6280 0.578517
\(405\) 40.8304 2.02888
\(406\) −21.4377 −1.06393
\(407\) −2.41517 −0.119716
\(408\) 28.1778 1.39501
\(409\) 34.9914 1.73021 0.865107 0.501588i \(-0.167251\pi\)
0.865107 + 0.501588i \(0.167251\pi\)
\(410\) 11.3608 0.561070
\(411\) 2.55591 0.126074
\(412\) −17.8147 −0.877669
\(413\) −23.6674 −1.16460
\(414\) 7.05287 0.346630
\(415\) −53.7370 −2.63785
\(416\) 29.3774 1.44034
\(417\) 27.6480 1.35393
\(418\) 1.03332 0.0505414
\(419\) 17.2771 0.844042 0.422021 0.906586i \(-0.361321\pi\)
0.422021 + 0.906586i \(0.361321\pi\)
\(420\) 23.6369 1.15336
\(421\) 9.62989 0.469332 0.234666 0.972076i \(-0.424600\pi\)
0.234666 + 0.972076i \(0.424600\pi\)
\(422\) −0.893036 −0.0434723
\(423\) 8.14397 0.395974
\(424\) 17.0811 0.829531
\(425\) −38.7710 −1.88067
\(426\) −3.32872 −0.161277
\(427\) 15.5236 0.751242
\(428\) 16.2519 0.785567
\(429\) −12.7061 −0.613454
\(430\) 23.8145 1.14844
\(431\) −25.2632 −1.21689 −0.608443 0.793598i \(-0.708205\pi\)
−0.608443 + 0.793598i \(0.708205\pi\)
\(432\) −0.510176 −0.0245459
\(433\) 12.6048 0.605747 0.302874 0.953031i \(-0.402054\pi\)
0.302874 + 0.953031i \(0.402054\pi\)
\(434\) −3.75331 −0.180164
\(435\) −70.3941 −3.37514
\(436\) −14.5556 −0.697087
\(437\) −5.80291 −0.277591
\(438\) −10.2619 −0.490335
\(439\) −0.465354 −0.0222101 −0.0111051 0.999938i \(-0.503535\pi\)
−0.0111051 + 0.999938i \(0.503535\pi\)
\(440\) 12.0309 0.573553
\(441\) −0.403904 −0.0192335
\(442\) −22.1554 −1.05383
\(443\) 28.9952 1.37761 0.688803 0.724949i \(-0.258137\pi\)
0.688803 + 0.724949i \(0.258137\pi\)
\(444\) 5.24149 0.248750
\(445\) −5.45465 −0.258575
\(446\) −13.2567 −0.627724
\(447\) 5.77700 0.273243
\(448\) −13.6891 −0.646749
\(449\) −29.8561 −1.40900 −0.704498 0.709706i \(-0.748828\pi\)
−0.704498 + 0.709706i \(0.748828\pi\)
\(450\) −9.98777 −0.470828
\(451\) −4.04882 −0.190651
\(452\) 9.96189 0.468568
\(453\) 0.626606 0.0294405
\(454\) 20.5046 0.962328
\(455\) −49.4962 −2.32042
\(456\) −5.97239 −0.279683
\(457\) 35.5833 1.66452 0.832259 0.554387i \(-0.187047\pi\)
0.832259 + 0.554387i \(0.187047\pi\)
\(458\) 18.1327 0.847287
\(459\) −16.1486 −0.753753
\(460\) −25.3690 −1.18284
\(461\) −1.55313 −0.0723363 −0.0361681 0.999346i \(-0.511515\pi\)
−0.0361681 + 0.999346i \(0.511515\pi\)
\(462\) 5.58687 0.259925
\(463\) 1.73303 0.0805407 0.0402703 0.999189i \(-0.487178\pi\)
0.0402703 + 0.999189i \(0.487178\pi\)
\(464\) 1.38201 0.0641581
\(465\) −12.3246 −0.571539
\(466\) 2.85196 0.132114
\(467\) 28.7920 1.33233 0.666167 0.745802i \(-0.267934\pi\)
0.666167 + 0.745802i \(0.267934\pi\)
\(468\) 8.60565 0.397796
\(469\) −9.57752 −0.442249
\(470\) 19.4282 0.896155
\(471\) −32.4383 −1.49468
\(472\) 26.1436 1.20336
\(473\) −8.48714 −0.390239
\(474\) 8.25320 0.379082
\(475\) 8.21767 0.377052
\(476\) −14.6886 −0.673251
\(477\) 8.12849 0.372178
\(478\) −9.00668 −0.411956
\(479\) 6.72157 0.307117 0.153558 0.988140i \(-0.450927\pi\)
0.153558 + 0.988140i \(0.450927\pi\)
\(480\) −42.4160 −1.93602
\(481\) −10.9758 −0.500452
\(482\) 7.56049 0.344371
\(483\) −31.3747 −1.42760
\(484\) 11.6174 0.528064
\(485\) −37.1091 −1.68504
\(486\) −11.7744 −0.534097
\(487\) −28.1002 −1.27334 −0.636671 0.771136i \(-0.719689\pi\)
−0.636671 + 0.771136i \(0.719689\pi\)
\(488\) −17.1478 −0.776244
\(489\) 11.2496 0.508726
\(490\) −0.963549 −0.0435287
\(491\) 5.59129 0.252332 0.126166 0.992009i \(-0.459733\pi\)
0.126166 + 0.992009i \(0.459733\pi\)
\(492\) 8.78687 0.396143
\(493\) 43.7447 1.97016
\(494\) 4.69593 0.211280
\(495\) 5.72525 0.257331
\(496\) 0.241962 0.0108644
\(497\) 4.62123 0.207291
\(498\) 27.5650 1.23522
\(499\) 23.4167 1.04828 0.524139 0.851633i \(-0.324387\pi\)
0.524139 + 0.851633i \(0.324387\pi\)
\(500\) 14.0669 0.629091
\(501\) −34.3197 −1.53329
\(502\) −7.22281 −0.322370
\(503\) −25.8124 −1.15092 −0.575459 0.817831i \(-0.695176\pi\)
−0.575459 + 0.817831i \(0.695176\pi\)
\(504\) −10.0774 −0.448884
\(505\) 35.1564 1.56444
\(506\) −5.99627 −0.266567
\(507\) −30.5949 −1.35877
\(508\) 13.3842 0.593826
\(509\) −23.5181 −1.04242 −0.521212 0.853427i \(-0.674520\pi\)
−0.521212 + 0.853427i \(0.674520\pi\)
\(510\) 31.9887 1.41648
\(511\) 14.2466 0.630231
\(512\) 1.68530 0.0744804
\(513\) 3.42276 0.151119
\(514\) 16.8807 0.744576
\(515\) −53.8613 −2.37341
\(516\) 18.4191 0.810854
\(517\) −6.92391 −0.304513
\(518\) 4.82607 0.212045
\(519\) −37.8935 −1.66334
\(520\) 54.6747 2.39764
\(521\) 16.2908 0.713715 0.356858 0.934159i \(-0.383848\pi\)
0.356858 + 0.934159i \(0.383848\pi\)
\(522\) 11.2690 0.493233
\(523\) 4.22810 0.184882 0.0924410 0.995718i \(-0.470533\pi\)
0.0924410 + 0.995718i \(0.470533\pi\)
\(524\) 13.0870 0.571710
\(525\) 44.4306 1.93911
\(526\) −0.356147 −0.0155288
\(527\) 7.65881 0.333623
\(528\) −0.360165 −0.0156742
\(529\) 10.6738 0.464078
\(530\) 19.3912 0.842302
\(531\) 12.4411 0.539899
\(532\) 3.11331 0.134979
\(533\) −18.3999 −0.796987
\(534\) 2.79802 0.121082
\(535\) 49.1363 2.12435
\(536\) 10.5796 0.456967
\(537\) 13.2130 0.570182
\(538\) −9.48026 −0.408723
\(539\) 0.343394 0.0147910
\(540\) 14.9635 0.643928
\(541\) 13.4238 0.577133 0.288567 0.957460i \(-0.406821\pi\)
0.288567 + 0.957460i \(0.406821\pi\)
\(542\) −12.2863 −0.527741
\(543\) 15.5456 0.667126
\(544\) 26.3584 1.13011
\(545\) −44.0076 −1.88508
\(546\) 25.3896 1.08657
\(547\) 17.8070 0.761371 0.380685 0.924705i \(-0.375688\pi\)
0.380685 + 0.924705i \(0.375688\pi\)
\(548\) 1.47175 0.0628699
\(549\) −8.16023 −0.348270
\(550\) 8.49148 0.362078
\(551\) −9.27186 −0.394995
\(552\) 34.6573 1.47511
\(553\) −11.4579 −0.487238
\(554\) 13.0085 0.552679
\(555\) 15.8472 0.672675
\(556\) 15.9203 0.675171
\(557\) −41.7680 −1.76977 −0.884884 0.465812i \(-0.845762\pi\)
−0.884884 + 0.465812i \(0.845762\pi\)
\(558\) 1.97298 0.0835230
\(559\) −38.5699 −1.63133
\(560\) −1.40302 −0.0592883
\(561\) −11.4003 −0.481321
\(562\) −16.5800 −0.699385
\(563\) 23.8677 1.00590 0.502952 0.864314i \(-0.332247\pi\)
0.502952 + 0.864314i \(0.332247\pi\)
\(564\) 15.0265 0.632729
\(565\) 30.1189 1.26711
\(566\) −2.95772 −0.124322
\(567\) 29.0769 1.22111
\(568\) −5.10472 −0.214190
\(569\) 38.1261 1.59833 0.799165 0.601111i \(-0.205275\pi\)
0.799165 + 0.601111i \(0.205275\pi\)
\(570\) −6.78014 −0.283989
\(571\) −9.87638 −0.413314 −0.206657 0.978413i \(-0.566258\pi\)
−0.206657 + 0.978413i \(0.566258\pi\)
\(572\) −7.31642 −0.305915
\(573\) 50.8220 2.12312
\(574\) 8.09046 0.337689
\(575\) −47.6864 −1.98866
\(576\) 7.19588 0.299828
\(577\) −23.6347 −0.983924 −0.491962 0.870617i \(-0.663720\pi\)
−0.491962 + 0.870617i \(0.663720\pi\)
\(578\) −4.69699 −0.195369
\(579\) −51.1605 −2.12616
\(580\) −40.5345 −1.68310
\(581\) −38.2682 −1.58763
\(582\) 19.0355 0.789046
\(583\) −6.91075 −0.286214
\(584\) −15.7371 −0.651206
\(585\) 26.0184 1.07573
\(586\) −4.85797 −0.200681
\(587\) −4.82609 −0.199194 −0.0995971 0.995028i \(-0.531755\pi\)
−0.0995971 + 0.995028i \(0.531755\pi\)
\(588\) −0.745246 −0.0307334
\(589\) −1.62332 −0.0668876
\(590\) 29.6794 1.22188
\(591\) −6.83390 −0.281109
\(592\) −0.311119 −0.0127869
\(593\) −18.4588 −0.758011 −0.379006 0.925394i \(-0.623734\pi\)
−0.379006 + 0.925394i \(0.623734\pi\)
\(594\) 3.53681 0.145117
\(595\) −44.4097 −1.82062
\(596\) 3.32653 0.136260
\(597\) 43.2896 1.77173
\(598\) −27.2501 −1.11434
\(599\) −38.4768 −1.57212 −0.786060 0.618150i \(-0.787883\pi\)
−0.786060 + 0.618150i \(0.787883\pi\)
\(600\) −49.0791 −2.00365
\(601\) 6.43726 0.262581 0.131291 0.991344i \(-0.458088\pi\)
0.131291 + 0.991344i \(0.458088\pi\)
\(602\) 16.9592 0.691207
\(603\) 5.03457 0.205023
\(604\) 0.360813 0.0146813
\(605\) 35.1242 1.42800
\(606\) −18.0338 −0.732574
\(607\) 3.21905 0.130657 0.0653285 0.997864i \(-0.479190\pi\)
0.0653285 + 0.997864i \(0.479190\pi\)
\(608\) −5.58676 −0.226573
\(609\) −50.1304 −2.03138
\(610\) −19.4670 −0.788194
\(611\) −31.4658 −1.27297
\(612\) 7.72128 0.312114
\(613\) 0.795323 0.0321228 0.0160614 0.999871i \(-0.494887\pi\)
0.0160614 + 0.999871i \(0.494887\pi\)
\(614\) −20.5746 −0.830322
\(615\) 26.5663 1.07126
\(616\) 8.56770 0.345202
\(617\) −9.50252 −0.382557 −0.191279 0.981536i \(-0.561263\pi\)
−0.191279 + 0.981536i \(0.561263\pi\)
\(618\) 27.6287 1.11139
\(619\) 14.3260 0.575811 0.287906 0.957659i \(-0.407041\pi\)
0.287906 + 0.957659i \(0.407041\pi\)
\(620\) −7.09677 −0.285013
\(621\) −19.8620 −0.797035
\(622\) 6.25009 0.250606
\(623\) −3.88446 −0.155628
\(624\) −1.63677 −0.0655233
\(625\) 1.44170 0.0576681
\(626\) 20.7410 0.828978
\(627\) 2.41634 0.0964993
\(628\) −18.6787 −0.745360
\(629\) −9.84784 −0.392659
\(630\) −11.4403 −0.455794
\(631\) 17.7905 0.708229 0.354114 0.935202i \(-0.384782\pi\)
0.354114 + 0.935202i \(0.384782\pi\)
\(632\) 12.6566 0.503453
\(633\) −2.08830 −0.0830023
\(634\) −9.13511 −0.362802
\(635\) 40.4658 1.60584
\(636\) 14.9979 0.594706
\(637\) 1.56056 0.0618316
\(638\) −9.58081 −0.379308
\(639\) −2.42922 −0.0960985
\(640\) −23.4562 −0.927187
\(641\) −12.6727 −0.500542 −0.250271 0.968176i \(-0.580520\pi\)
−0.250271 + 0.968176i \(0.580520\pi\)
\(642\) −25.2049 −0.994760
\(643\) 19.4811 0.768260 0.384130 0.923279i \(-0.374501\pi\)
0.384130 + 0.923279i \(0.374501\pi\)
\(644\) −18.0663 −0.711910
\(645\) 55.6884 2.19273
\(646\) 4.21335 0.165772
\(647\) −37.4744 −1.47327 −0.736636 0.676289i \(-0.763587\pi\)
−0.736636 + 0.676289i \(0.763587\pi\)
\(648\) −32.1190 −1.26175
\(649\) −10.5773 −0.415196
\(650\) 38.5896 1.51361
\(651\) −8.77681 −0.343990
\(652\) 6.47779 0.253690
\(653\) 16.3762 0.640849 0.320425 0.947274i \(-0.396174\pi\)
0.320425 + 0.947274i \(0.396174\pi\)
\(654\) 22.5741 0.882718
\(655\) 39.5675 1.54603
\(656\) −0.521562 −0.0203636
\(657\) −7.48893 −0.292171
\(658\) 13.8355 0.539366
\(659\) 19.8667 0.773897 0.386948 0.922101i \(-0.373529\pi\)
0.386948 + 0.922101i \(0.373529\pi\)
\(660\) 10.5637 0.411191
\(661\) −27.5440 −1.07134 −0.535669 0.844428i \(-0.679941\pi\)
−0.535669 + 0.844428i \(0.679941\pi\)
\(662\) −3.98887 −0.155032
\(663\) −51.8087 −2.01208
\(664\) 42.2720 1.64047
\(665\) 9.41281 0.365013
\(666\) −2.53689 −0.0983027
\(667\) 53.8038 2.08329
\(668\) −19.7620 −0.764617
\(669\) −30.9998 −1.19852
\(670\) 12.0104 0.464002
\(671\) 6.93773 0.267828
\(672\) −30.2061 −1.16522
\(673\) 37.5288 1.44663 0.723315 0.690518i \(-0.242617\pi\)
0.723315 + 0.690518i \(0.242617\pi\)
\(674\) 26.3799 1.01612
\(675\) 28.1271 1.08261
\(676\) −17.6172 −0.677585
\(677\) −3.47250 −0.133459 −0.0667296 0.997771i \(-0.521256\pi\)
−0.0667296 + 0.997771i \(0.521256\pi\)
\(678\) −15.4498 −0.593346
\(679\) −26.4268 −1.01417
\(680\) 49.0560 1.88121
\(681\) 47.9484 1.83738
\(682\) −1.67741 −0.0642312
\(683\) 1.73573 0.0664160 0.0332080 0.999448i \(-0.489428\pi\)
0.0332080 + 0.999448i \(0.489428\pi\)
\(684\) −1.63656 −0.0625753
\(685\) 4.44970 0.170014
\(686\) −16.8710 −0.644139
\(687\) 42.4020 1.61774
\(688\) −1.09330 −0.0416816
\(689\) −31.4059 −1.19647
\(690\) 39.3445 1.49782
\(691\) 44.5147 1.69342 0.846709 0.532056i \(-0.178580\pi\)
0.846709 + 0.532056i \(0.178580\pi\)
\(692\) −21.8199 −0.829468
\(693\) 4.07717 0.154879
\(694\) 24.5347 0.931326
\(695\) 48.1336 1.82581
\(696\) 55.3752 2.09899
\(697\) −16.5090 −0.625323
\(698\) −10.4027 −0.393747
\(699\) 6.66908 0.252248
\(700\) 25.5841 0.966989
\(701\) −18.1133 −0.684128 −0.342064 0.939677i \(-0.611126\pi\)
−0.342064 + 0.939677i \(0.611126\pi\)
\(702\) 16.0731 0.606639
\(703\) 2.08729 0.0787236
\(704\) −6.11785 −0.230575
\(705\) 45.4313 1.71104
\(706\) −32.8627 −1.23681
\(707\) 25.0362 0.941583
\(708\) 22.9552 0.862709
\(709\) 32.6750 1.22714 0.613568 0.789642i \(-0.289734\pi\)
0.613568 + 0.789642i \(0.289734\pi\)
\(710\) −5.79512 −0.217487
\(711\) 6.02299 0.225880
\(712\) 4.29087 0.160807
\(713\) 9.41996 0.352780
\(714\) 22.7804 0.852535
\(715\) −22.1206 −0.827262
\(716\) 7.60833 0.284336
\(717\) −21.0614 −0.786553
\(718\) 23.1416 0.863635
\(719\) −42.2722 −1.57649 −0.788243 0.615364i \(-0.789009\pi\)
−0.788243 + 0.615364i \(0.789009\pi\)
\(720\) 0.737517 0.0274856
\(721\) −38.3567 −1.42848
\(722\) −0.893036 −0.0332354
\(723\) 17.6796 0.657512
\(724\) 8.95151 0.332680
\(725\) −76.1931 −2.82974
\(726\) −18.0173 −0.668685
\(727\) 25.3945 0.941830 0.470915 0.882178i \(-0.343924\pi\)
0.470915 + 0.882178i \(0.343924\pi\)
\(728\) 38.9360 1.44306
\(729\) 6.15853 0.228094
\(730\) −17.8655 −0.661232
\(731\) −34.6062 −1.27996
\(732\) −15.0565 −0.556504
\(733\) −41.9551 −1.54965 −0.774823 0.632179i \(-0.782161\pi\)
−0.774823 + 0.632179i \(0.782161\pi\)
\(734\) −23.0543 −0.850949
\(735\) −2.25318 −0.0831099
\(736\) 32.4195 1.19500
\(737\) −4.28033 −0.157668
\(738\) −4.25287 −0.156550
\(739\) 6.15000 0.226232 0.113116 0.993582i \(-0.463917\pi\)
0.113116 + 0.993582i \(0.463917\pi\)
\(740\) 9.12515 0.335447
\(741\) 10.9811 0.403400
\(742\) 13.8092 0.506953
\(743\) −23.6412 −0.867310 −0.433655 0.901079i \(-0.642776\pi\)
−0.433655 + 0.901079i \(0.642776\pi\)
\(744\) 9.69508 0.355439
\(745\) 10.0575 0.368477
\(746\) 8.62391 0.315744
\(747\) 20.1163 0.736015
\(748\) −6.56454 −0.240024
\(749\) 34.9918 1.27857
\(750\) −21.8162 −0.796615
\(751\) −35.8648 −1.30872 −0.654362 0.756182i \(-0.727063\pi\)
−0.654362 + 0.756182i \(0.727063\pi\)
\(752\) −0.891926 −0.0325252
\(753\) −16.8900 −0.615505
\(754\) −43.5400 −1.58563
\(755\) 1.09089 0.0397015
\(756\) 10.6561 0.387559
\(757\) 21.8139 0.792839 0.396419 0.918070i \(-0.370253\pi\)
0.396419 + 0.918070i \(0.370253\pi\)
\(758\) −21.9515 −0.797315
\(759\) −14.0218 −0.508959
\(760\) −10.3976 −0.377161
\(761\) −24.2093 −0.877586 −0.438793 0.898588i \(-0.644594\pi\)
−0.438793 + 0.898588i \(0.644594\pi\)
\(762\) −20.7574 −0.751960
\(763\) −31.3395 −1.13457
\(764\) 29.2644 1.05875
\(765\) 23.3446 0.844026
\(766\) 16.1589 0.583846
\(767\) −48.0686 −1.73566
\(768\) 34.1149 1.23102
\(769\) −35.2891 −1.27256 −0.636279 0.771459i \(-0.719527\pi\)
−0.636279 + 0.771459i \(0.719527\pi\)
\(770\) 9.72645 0.350517
\(771\) 39.4742 1.42163
\(772\) −29.4593 −1.06026
\(773\) 9.97739 0.358862 0.179431 0.983771i \(-0.442574\pi\)
0.179431 + 0.983771i \(0.442574\pi\)
\(774\) −8.91487 −0.320439
\(775\) −13.3399 −0.479182
\(776\) 29.1917 1.04792
\(777\) 11.2854 0.404861
\(778\) 4.19175 0.150281
\(779\) 3.49915 0.125370
\(780\) 48.0067 1.71892
\(781\) 2.06529 0.0739020
\(782\) −24.4497 −0.874320
\(783\) −31.7354 −1.13413
\(784\) 0.0442355 0.00157984
\(785\) −56.4733 −2.01562
\(786\) −20.2965 −0.723953
\(787\) −49.6948 −1.77143 −0.885714 0.464232i \(-0.846330\pi\)
−0.885714 + 0.464232i \(0.846330\pi\)
\(788\) −3.93511 −0.140183
\(789\) −0.832823 −0.0296493
\(790\) 14.3684 0.511204
\(791\) 21.4488 0.762633
\(792\) −4.50374 −0.160033
\(793\) 31.5286 1.11961
\(794\) −7.61361 −0.270197
\(795\) 45.3449 1.60822
\(796\) 24.9271 0.883518
\(797\) 38.0061 1.34625 0.673123 0.739531i \(-0.264952\pi\)
0.673123 + 0.739531i \(0.264952\pi\)
\(798\) −4.82840 −0.170923
\(799\) −28.2321 −0.998781
\(800\) −45.9101 −1.62317
\(801\) 2.04193 0.0721479
\(802\) −13.5291 −0.477729
\(803\) 6.36700 0.224686
\(804\) 9.28930 0.327608
\(805\) −54.6217 −1.92516
\(806\) −7.62298 −0.268508
\(807\) −22.1688 −0.780380
\(808\) −27.6556 −0.972920
\(809\) 2.22009 0.0780543 0.0390272 0.999238i \(-0.487574\pi\)
0.0390272 + 0.999238i \(0.487574\pi\)
\(810\) −36.4630 −1.28118
\(811\) 30.1952 1.06030 0.530149 0.847904i \(-0.322136\pi\)
0.530149 + 0.847904i \(0.322136\pi\)
\(812\) −28.8662 −1.01300
\(813\) −28.7305 −1.00762
\(814\) 2.15684 0.0755971
\(815\) 19.5850 0.686033
\(816\) −1.46857 −0.0514101
\(817\) 7.33492 0.256616
\(818\) −31.2486 −1.09258
\(819\) 18.5287 0.647446
\(820\) 15.2975 0.534211
\(821\) 17.2989 0.603737 0.301868 0.953350i \(-0.402390\pi\)
0.301868 + 0.953350i \(0.402390\pi\)
\(822\) −2.28252 −0.0796120
\(823\) 8.08683 0.281889 0.140945 0.990017i \(-0.454986\pi\)
0.140945 + 0.990017i \(0.454986\pi\)
\(824\) 42.3697 1.47602
\(825\) 19.8567 0.691320
\(826\) 21.1359 0.735411
\(827\) −12.2290 −0.425244 −0.212622 0.977135i \(-0.568200\pi\)
−0.212622 + 0.977135i \(0.568200\pi\)
\(828\) 9.49680 0.330036
\(829\) 53.1727 1.84676 0.923382 0.383883i \(-0.125413\pi\)
0.923382 + 0.383883i \(0.125413\pi\)
\(830\) 47.9891 1.66573
\(831\) 30.4194 1.05524
\(832\) −27.8026 −0.963882
\(833\) 1.40019 0.0485136
\(834\) −24.6906 −0.854967
\(835\) −59.7488 −2.06769
\(836\) 1.39138 0.0481219
\(837\) −5.55623 −0.192051
\(838\) −15.4291 −0.532989
\(839\) 17.6150 0.608137 0.304068 0.952650i \(-0.401655\pi\)
0.304068 + 0.952650i \(0.401655\pi\)
\(840\) −56.2170 −1.93967
\(841\) 56.9675 1.96440
\(842\) −8.59984 −0.296370
\(843\) −38.7710 −1.33534
\(844\) −1.20249 −0.0413913
\(845\) −53.2641 −1.83234
\(846\) −7.27286 −0.250046
\(847\) 25.0133 0.859467
\(848\) −0.890231 −0.0305707
\(849\) −6.91641 −0.237370
\(850\) 34.6239 1.18759
\(851\) −12.1124 −0.415206
\(852\) −4.48217 −0.153557
\(853\) 15.7499 0.539265 0.269632 0.962963i \(-0.413098\pi\)
0.269632 + 0.962963i \(0.413098\pi\)
\(854\) −13.8632 −0.474388
\(855\) −4.94798 −0.169217
\(856\) −38.6528 −1.32113
\(857\) −20.1489 −0.688272 −0.344136 0.938920i \(-0.611828\pi\)
−0.344136 + 0.938920i \(0.611828\pi\)
\(858\) 11.3470 0.387379
\(859\) 25.0443 0.854501 0.427250 0.904133i \(-0.359482\pi\)
0.427250 + 0.904133i \(0.359482\pi\)
\(860\) 32.0666 1.09346
\(861\) 18.9189 0.644755
\(862\) 22.5609 0.768429
\(863\) −3.99470 −0.135981 −0.0679906 0.997686i \(-0.521659\pi\)
−0.0679906 + 0.997686i \(0.521659\pi\)
\(864\) −19.1222 −0.650549
\(865\) −65.9705 −2.24306
\(866\) −11.2565 −0.382513
\(867\) −10.9835 −0.373021
\(868\) −5.05388 −0.171540
\(869\) −5.12068 −0.173707
\(870\) 62.8645 2.13131
\(871\) −19.4520 −0.659105
\(872\) 34.6183 1.17232
\(873\) 13.8916 0.470161
\(874\) 5.18221 0.175291
\(875\) 30.2873 1.02390
\(876\) −13.8179 −0.466862
\(877\) 7.25772 0.245076 0.122538 0.992464i \(-0.460897\pi\)
0.122538 + 0.992464i \(0.460897\pi\)
\(878\) 0.415578 0.0140251
\(879\) −11.3600 −0.383163
\(880\) −0.627028 −0.0211371
\(881\) 34.7122 1.16948 0.584742 0.811219i \(-0.301196\pi\)
0.584742 + 0.811219i \(0.301196\pi\)
\(882\) 0.360701 0.0121454
\(883\) 34.6895 1.16740 0.583698 0.811971i \(-0.301605\pi\)
0.583698 + 0.811971i \(0.301605\pi\)
\(884\) −29.8326 −1.00338
\(885\) 69.4030 2.33296
\(886\) −25.8938 −0.869919
\(887\) 19.9176 0.668769 0.334384 0.942437i \(-0.391472\pi\)
0.334384 + 0.942437i \(0.391472\pi\)
\(888\) −12.4661 −0.418335
\(889\) 28.8173 0.966500
\(890\) 4.87120 0.163283
\(891\) 12.9949 0.435344
\(892\) −17.8504 −0.597674
\(893\) 5.98392 0.200244
\(894\) −5.15907 −0.172545
\(895\) 23.0031 0.768909
\(896\) −16.7040 −0.558043
\(897\) −63.7222 −2.12762
\(898\) 26.6626 0.889741
\(899\) 15.0512 0.501985
\(900\) −13.4487 −0.448289
\(901\) −28.1785 −0.938761
\(902\) 3.61574 0.120391
\(903\) 39.6578 1.31973
\(904\) −23.6929 −0.788014
\(905\) 27.0641 0.899641
\(906\) −0.559582 −0.0185909
\(907\) 24.3288 0.807825 0.403913 0.914798i \(-0.367650\pi\)
0.403913 + 0.914798i \(0.367650\pi\)
\(908\) 27.6097 0.916261
\(909\) −13.1607 −0.436511
\(910\) 44.2019 1.46528
\(911\) −33.5883 −1.11283 −0.556416 0.830904i \(-0.687824\pi\)
−0.556416 + 0.830904i \(0.687824\pi\)
\(912\) 0.311269 0.0103071
\(913\) −17.1026 −0.566014
\(914\) −31.7772 −1.05110
\(915\) −45.5220 −1.50491
\(916\) 24.4160 0.806727
\(917\) 28.1775 0.930504
\(918\) 14.4213 0.475974
\(919\) −41.9286 −1.38310 −0.691549 0.722329i \(-0.743071\pi\)
−0.691549 + 0.722329i \(0.743071\pi\)
\(920\) 60.3365 1.98923
\(921\) −48.1120 −1.58534
\(922\) 1.38700 0.0456783
\(923\) 9.38574 0.308936
\(924\) 7.52281 0.247482
\(925\) 17.1526 0.563975
\(926\) −1.54766 −0.0508592
\(927\) 20.1628 0.662232
\(928\) 51.7997 1.70041
\(929\) −38.9179 −1.27685 −0.638427 0.769683i \(-0.720414\pi\)
−0.638427 + 0.769683i \(0.720414\pi\)
\(930\) 11.0063 0.360911
\(931\) −0.296775 −0.00972641
\(932\) 3.84021 0.125790
\(933\) 14.6153 0.478485
\(934\) −25.7123 −0.841332
\(935\) −19.8473 −0.649077
\(936\) −20.4673 −0.668994
\(937\) 58.4744 1.91027 0.955137 0.296164i \(-0.0957076\pi\)
0.955137 + 0.296164i \(0.0957076\pi\)
\(938\) 8.55307 0.279268
\(939\) 48.5013 1.58278
\(940\) 26.1603 0.853255
\(941\) 11.2067 0.365328 0.182664 0.983175i \(-0.441528\pi\)
0.182664 + 0.983175i \(0.441528\pi\)
\(942\) 28.9686 0.943847
\(943\) −20.3053 −0.661230
\(944\) −1.36255 −0.0443472
\(945\) 32.2178 1.04805
\(946\) 7.57932 0.246425
\(947\) −44.4900 −1.44573 −0.722866 0.690989i \(-0.757175\pi\)
−0.722866 + 0.690989i \(0.757175\pi\)
\(948\) 11.1131 0.360935
\(949\) 28.9349 0.939265
\(950\) −7.33867 −0.238098
\(951\) −21.3617 −0.692702
\(952\) 34.9347 1.13224
\(953\) −57.2567 −1.85473 −0.927363 0.374164i \(-0.877930\pi\)
−0.927363 + 0.374164i \(0.877930\pi\)
\(954\) −7.25904 −0.235020
\(955\) 88.4784 2.86309
\(956\) −12.1276 −0.392235
\(957\) −22.4040 −0.724217
\(958\) −6.00261 −0.193936
\(959\) 3.16880 0.102326
\(960\) 40.1423 1.29559
\(961\) −28.3648 −0.914995
\(962\) 9.80176 0.316021
\(963\) −18.3940 −0.592738
\(964\) 10.1803 0.327886
\(965\) −89.0676 −2.86719
\(966\) 28.0188 0.901489
\(967\) −11.2747 −0.362571 −0.181286 0.983431i \(-0.558026\pi\)
−0.181286 + 0.983431i \(0.558026\pi\)
\(968\) −27.6303 −0.888071
\(969\) 9.85259 0.316511
\(970\) 33.1398 1.06405
\(971\) 10.1284 0.325035 0.162517 0.986706i \(-0.448039\pi\)
0.162517 + 0.986706i \(0.448039\pi\)
\(972\) −15.8544 −0.508530
\(973\) 34.2778 1.09890
\(974\) 25.0945 0.804080
\(975\) 90.2388 2.88995
\(976\) 0.893707 0.0286069
\(977\) 17.5695 0.562097 0.281049 0.959694i \(-0.409318\pi\)
0.281049 + 0.959694i \(0.409318\pi\)
\(978\) −10.0463 −0.321246
\(979\) −1.73602 −0.0554835
\(980\) −1.29743 −0.0414450
\(981\) 16.4741 0.525976
\(982\) −4.99323 −0.159340
\(983\) 19.2044 0.612524 0.306262 0.951947i \(-0.400922\pi\)
0.306262 + 0.951947i \(0.400922\pi\)
\(984\) −20.8983 −0.666213
\(985\) −11.8975 −0.379085
\(986\) −39.0656 −1.24410
\(987\) 32.3533 1.02982
\(988\) 6.32314 0.201166
\(989\) −42.5639 −1.35345
\(990\) −5.11285 −0.162497
\(991\) −9.04212 −0.287232 −0.143616 0.989633i \(-0.545873\pi\)
−0.143616 + 0.989633i \(0.545873\pi\)
\(992\) 9.06908 0.287944
\(993\) −9.32765 −0.296004
\(994\) −4.12693 −0.130898
\(995\) 75.3649 2.38923
\(996\) 37.1166 1.17609
\(997\) 44.1747 1.39903 0.699513 0.714620i \(-0.253400\pi\)
0.699513 + 0.714620i \(0.253400\pi\)
\(998\) −20.9120 −0.661958
\(999\) 7.14429 0.226035
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4009.2.a.c.1.31 71
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4009.2.a.c.1.31 71 1.1 even 1 trivial